Download Mathcad - CO Rotational States.

Rotational States of the Carbon Monoxide
David M. Hanson
State University of New York at Stony Brook
Stony Brook, NY 11794-3400
[email protected]
and
Theresa Julia Zielinski
Monmouth University
West Long Branch NJ 07764-1898
[email protected]
© Copyright David M. Hanson and Theresa Julia Zielinski, 2002. All rights reserved. You
are welcome to use this document in your own classes but commercial use is not
allowed without the permission of the author.
Introduction: The magnitude of the absorption coefficient or the intensity of an absorption peak
in rotational spectroscopy is due to several factors First is the relative populations of the
rotational energy levels as given by the Boltzman constant. Second is the degeneracy of each
of the rotational energy levels. Third is the contribution of the partition function and finally the
transition moment for that transition. This activity permits you to examine each of these
contributions individually and then all together.
Prerequisites: You should be able to:
1. compute the degeneracy of rotational energy levels and then the relative population of energy
levels using the Boltzman distribution,
2. compute the frequencies of allowed rotational transitions,
3. use transition moments to improve the quality of the intensity of rotational spectrum peaks,
4. use centrifugal distortion as a correction to the calculation of the frequency of a rotational
transition,
5. Use Mathcad with skill to create new functions and plots of those functions.
Learning Objectives
1. Demonstrate that the spacing between spectral lines is nearly equal. Compute the percent
difference between the space between the first two spectral lines and the last two spectral lines.
2. Describe the effect of the Boltzman distribution on the shape of the rotation spectrum
of a diatomic molecule.
3. Distinguish between the effects of degeneracy of states and transition moments with respect
to the amplitude of a peak in the rotational spectrum.
4. Justify the need to consider centrifugal distortion and the partition function in generating a
spectrum for a rotating molecule.
Performance Criteria
1. High quality demonstration of mastery of the learning objectives.
2. High quality responses to the questions, exercises, and problems below.
3. High quality definitions of the terms transition moment and centrifugal distortion.
4. Success in analyzing other spectra using the principles of Quantum Mechanics derived
for the rigid rotor.
Hanson & Zielinski
CO Rotational States.mcd
page 1
Background Information for the Activity
The fraction of the microwave power transmitted through a cell, P/P 0 , which is called the
transmittance, T, decays exponentially with the length of the cell, d, and the magnitude of the
absorption coefficient, γ, which varies with the frequency of the radiation, v. SI units are used in
this analysis so the absorption coefficient has units of m -1.
T=
P
P0
− γd
=e
An absorption spectrum consists of a plot of the absorption coefficient as a function of the
microwave frequency, γ(v). Such a spectrum usually is obtained experimentally as the absorbance
vs frequency since the absorbance is log(P 0 /P), which equals γd/2.303.
The absorption coefficient, γ(v), is a sum of the absorption coefficients, γJ(v), associated
with the transition from each initial rotational energy level J to the final level J+1. Each γJ(v) can be
written as a product of several factors that are defined below.
C = all the constants that appear at various steps in the derivation.
N = the number of molecules per m 3 in the absorption cell.
DJ = the square of the transition dipole moment for the transition between energy levels, depends
on the permanent dipole moment of the molecule µ and the rotational quantum number J of the
initial level.
PJ = a factor accounting for the population difference between the initial and final energy levels.
Fn = the fraction of molecules in vibrational state n.
FJ = the fraction of molecules in rotational energy level J. Note that Fn FJ is the fraction of molecules
in vibrational level n and rotational level J.
L(v,J) = a lineshape function that describes how the absorption coefficient γJ(v) associated with a
transition from level J to J+1 depends on the microwave frequency.
Review of concepts
For a molecule in the nth vibrational state, the rotational energy of the J rotational level is given by:
EJ
h
Hanson & Zielinski
2
2
= J ( J + 1 ) ⋅ Bn − J ( J + 1 ) ⋅ Dc
CO Rotational States.mcd
(1)
page 2
where Bn is the rotational constant for a molecule in the n th vibrational state and Dc is the
centrifugal stretching constant.


Bn = Be −  n +
1
2
⋅α
(2)
where Be is the rotational constant and α is the rotation-vibration interaction constant.
We will construct the rotation spectrum for CO with a function known as a Lorentian. The Lorentian
function is a standard function used to model spectral peaks.
L ( v , J ) :=
∆ν

 ⋅ Emax
2
2
 ( v − ν J) + ∆ν 
A Lorentian Function
(3)
Notice the structure of the function. Emax is the amplitude maximum for a peak in a spectrum. This
peak is centered at ν J in a spectrum that has a range v. ∆ν is the width of the peak at half height.
Each peak in a spectrum has a function like this to describe it. The full spectrum is obtained by
plotting the Lorentian function as a function of v and J where J is the quantum number for the origin
of the rotational transition giving rise to the peak.
Exercise 1. Using only equation 3 sketch a Lorentian function from the information given
here. Label all parts of the sketch.
With these basic equations we can proceed to produce a rotational spectrum.
First, however, we need to do some preliminary work to prepare for drawing the spectrum.
Define some of the constants needed in this activity.
8 m
c := 2.99792458 ⋅ 10 ⋅
sec
− 34
h := 6.6260755 ⋅ 10
Speed of light
⋅ joule ⋅ sec Plank's constant
− 23 joule
k b := 1.380658 ⋅ 10
Hanson & Zielinski
⋅
K
Boltzmann's constant
CO Rotational States.mcd
page 3
Compute Some Needed Values
Reduced Mass
Carbon Mass
mC :=
Oxygen Mass
0.012000
6.023 ⋅ 10
23
⋅ kg
.0159949
23
⋅ kg
− 10
Re := 1.1282 ⋅ 10
⋅m
6.023 ⋅ 10
− 26
mC = 1.992 × 10
µ :=
mO :=
Equilibrium Bond Length
mO = 2.656 × 10
kg
mC ⋅ mO
− 26
kg
atomic masses
Reduced Mass
mC + mO
µ = 1.138 × 10
− 26
kg
Momnent of Inertia
I := µ ⋅ Re
2
− 46
I = 1.449 × 10
2
kg m
Rotational Constant
h
Be :=
2
8 ⋅π ⋅I ⋅c
-1
Be = 193.198 m
Note the Units on this constant.
T := 40 ⋅ K
Set the Temperature
Exercise 2. Since the temperature is very low, in which vibrational state will all
of the molecules be found? What values are allowed for the vibrational
quantum numbers?
Compute B0
α := 17.48 ⋅ m


−1


B0 :=  Be −  n +
n := 0
1
2


⋅α
-1
B0 = 184.458 m
Hanson & Zielinski
CO Rotational States.mcd
page 4
Compute the Energy Level Energies
J := 0 , 1 .. 9
Define the range of rotational energy levels to study.
2
EJ :=
h
2
⋅ J ⋅ ( J + 1)
Use this equation to compute the 10 energy levels ignoring
centrifugal stretching.
8 ⋅π ⋅I
Exercise 3. Ask Mathcad to show these in the space to the right. What is the
magnitude of the energy? What are the units?
Compute the Delta E for each transition given that the allowed transitions are J = +1
or -1. For absorption we use +1.
k := 1 .. 9
DeltaEk := Ek − Ek− 1
alternatively
ν J := B0 ⋅ 2 ⋅ ( J + 1 ) ignoring centrifugal stretching
Exercise 4. Display the J, EJ, DeltaEk, and ν J . Describe you observations on the results.
What region of the electromagnetic spectrum does the data tell you to find the spectrum?
Exercie 4. Verify that vJ := B0 ⋅ 2 ⋅ ( J + 1 ) is the correct expression for the difference between
adjacent energy levels for the CO rotor when centrifugal stretching is ignored.
Exercise 5. Verify that DeltaEkk is the same as ν J.
Exercise 6. Convert the frequencies, ν J , given above into Hz and MHz.
Build the Spectrum
Now we will use our knowledge about Rotational States to construct a spectrum for CO in the
microwave region. Each peak in the spectrum results from one of the transitions we calculated
above. In this activity you will add successive corrections to the spectrum in order to understand why
the rotational spectrum has the shape we see experimentally. Specifically you will adjust the
amplitude of each peak to reflect the following:
•
•
•
•
•
•
the degeneracy of the energy levels,
the effect of temperature,
the effect of including rotational and vibrational partition functions,
the population of the energy level leading to the transition,
centrifugal stretching,
the magnitude of the transition moment for that transition.
We will use the Lorentzian to construct the spectrum.
Hanson & Zielinski
CO Rotational States.mcd
page 5
Define the frequency range as 100,000 to 1,200,000 MHz. This will accomodate the ten
lines we will calculate each time we adjust the amplitude as we analyze the spectrum of CO.
5
5
6
v := 1.00 ⋅ 10 ⋅ MHz , 1.01 ⋅ 10 ⋅ MHz .. 1.20 ⋅ 10 ⋅ MHz
Defines the width of each absorption peak.
∆ν := 7500 ⋅ MHz
νν J := Be ⋅ c ⋅ 2 ⋅ ( J + 1 )
L ( v , J ) :=
∆ν


2
2
 ( v − νν J) + ∆ν 
A Lorentian Function
For practice here we are saying that all the absorption coefficient intensities
are the same. This is not true but it permits us to explore the contribution of
∆ν in the spectrum.
Γ J := 1
γ ( v) :=
Rather than redefine a variable name we will use a new variable
name for frequency.
∑ (Γ J ⋅ L ( v , J) )
J
1 .10
10
5 .10
11
This gives the sum of all peaks defined by the Lorentian function.
When this is plotted as a function of the frequency range we get a
spectrum.
γ ( v)
0
0
5
2 .10
5
4 .10
5
6 .10
v
5
8 .10
6
1 .10
1.2 .10
6
MHz
Exercise 7. Systematically vary ∆ν and describe your observatons for the plot above for
lower and higher valuse of ∆ν. Do your changes in terms of 500 MHz steps?
• How does changing the width at half height affect the spectrum?
• What is the consequence of using B0 instead of Be? What is the significance of using
B0 instead of Be?
Hanson & Zielinski
CO Rotational States.mcd
page 6
You be the Spectroscopist
Below we list the essential components that lead to a rotational spectrum for CO. Use the data here to
create spectrum that includes each of the significant factors.
ε0 := 8.854187817 ⋅ 10
− 12 farad
⋅
m
κ := 4 ⋅ π ⋅ ε0
− 10
κ = 1.113 × 10
C := 8 ⋅
π
2
2
-1
sec coul kg
-3
m
2
3 ⋅κ ⋅c ⋅h
C = 1.191 × 10
36
C is the accumulation of all the constants that arise in the
derivation of the absorption coefficient Γ used below.
-2
coul
13
ν n := 6.4252 ⋅ 10
−1
⋅ sec
-1
Zero vibrational level for Carbon Monoxide.
B0 = 184.458 m
Rotational Constant for CO in the zero vibrational level.
Dc := 0.18358 ⋅ MHz
Cetrifugal stretching constant for carbon monoxide.
D := 3.33564 ⋅ 10
− 30
⋅ coul ⋅ m
DM := 0.11 ⋅ D
Dipole Moment of Carbon Monoxide in Debyes
− 31
DM = 3.669 × 10
2
DJ := DM ⋅
FJ :=
The definition of a Debye unit for dipole moments
m coul
( J + 1)
( 2 ⋅ J + 1)
 h ⋅ B0 ⋅ c  ⋅ exp  −h ⋅ B0 ⋅ J ⋅ c ⋅ ( J + 1) 
 ⋅T


kb ⋅ T
 kb 


DG J := 2 ⋅ J + 1
This is the fraction of molecules in a particular
energy level. It is obtained from the rotational
partition function.
This is the degeneracy of each energy level.
 −h ⋅ ν n 
Fn := 1 − exp 
 kb ⋅ T 
CNTJ := −4 ⋅ Dc ⋅ ( J + 1 )
Hanson & Zielinski
3
Centrifugal stretching contribution to the frequency of
a peak in the spectrum.
CO Rotational States.mcd
page 7
The frequency expression that you will modify in
your exploration of the intensities of the lines in the
rotational spectrum of CO
ννν J := Be ⋅ c ⋅ 2 ⋅ ( J + 1 ) + CNTJ
 −h ⋅ ννν J ⋅ 106 
PJ := 1 − exp
 kb ⋅ T


23
N :=
3.1 ⋅ 10
m
Number of molecules in one meter cubed of the sample.
3
ΓΓ J is the factor that you will ,modify to include the
ΓΓ J := C ⋅ N ⋅ FJ ⋅ Fn ⋅ DJ ⋅ DG J
degeneracy of the transition, the population of the origin
of the transition, and the transition moment of a transition.
All these factors affect the absorption intensity of peaks
observed by spectroscopists.
2
∆ν

 ⋅ v
L ( v , J ) := 
 ( v − ννν J) 2 + ∆ν 2  ννν J
γγγ ( v) :=
∑ (ΓΓ J ⋅L(v , J))
J
0.4
γγγ ( v) 0.2
0
0
2 .10
5
5
4 .10
5
6 .10
5
8 .10
6
1 .10
6
1.2 .10
v
MHz
Compute the difference between peaks.
I := 0 .. 8
MI := ννν I+1 − ννν I
Hanson & Zielinski
CO Rotational States.mcd
page 8
MI =
1.1583·105
MHz
1.1582·105
1.1581·105
1.1579·105
1.1577·105
1.1575·105
1.1571·105
1.1568·105
1.1564·105
Exercise 8. Your job as the spectroscopist is to study the effect of each term in Γ.
•
•
You do this by starting with ΓΓ equal to one.
Then add terms in the following order: C, N, F J , Fn, PJ , D J , and DG J .
•
•
•
Report the consequence and significance of each factor.
How do your results differ from the experimental peak intensities?
What else do you need to consider in order to reproduce the experimental spectrum.
Exercise 9. Modify this document to generate the rotational spectrum for HCl .
References
C.H. Townes and A.L. Schalow, Microwave Spectroscopy (Dover Publications, New York, 1975
and McGraw-Hill, 1955) pp. 19 - 24 and pp. 336 - 342.
W. Gordy, R.L. Cook, Microwave Molecular Spectra (John Wiley & Sons, New York, 1984) p. 75.
Hanson & Zielinski
CO Rotational States.mcd
page 9